The radii of circular orbits of two satellite `A` and `B` of the earth are `4R` and `R`, respectively. If the speed of satellite `A` is `3v`, then the speed of satellite `B` will be

Options

A. `3v//4`
B. `6v`
C. `12v`
D. `3v//2`

Answer

To find the speed of satellite B, we will use the relationship between the orbital speed of a satellite and the radius of its orbit. The orbital speed \( v \) of a satellite in a circular orbit is given by the formula:

\[ v = \sqrt{\frac{GM}{r}} \]

where:
- \( G \) is the gravitational constant,
- \( M \) is the mass of the Earth (or the planet),
- \( r \) is the radius of the orbit.

### Step 1: Identify the given values
- For satellite A:
- Radius \( R_A = 4R \)
- Speed \( v_A = 3v \)

- For satellite B:
- Radius \( R_B = R \)
- Speed \( v_B = ? \)

### Step 2: Write the expression for the orbital speeds
Using the formula for orbital speed, we can write:

\[ v_A = \sqrt{\frac{GM}{R_A}} \quad \text{and} \quad v_B = \sqrt{\frac{GM}{R_B}} \]

### Step 3: Substitute the known values
Substituting the values of \( R_A \) and \( R_B \):

\[ v_A = \sqrt{\frac{GM}{4R}} \quad \text{and} \quad v_B = \sqrt{\frac{GM}{R}} \]

### Step 4: Relate the speeds of the satellites
From the expression for \( v_A \):

\[ v_A = \sqrt{\frac{GM}{4R}} = \frac{1}{2} \sqrt{\frac{GM}{R}} = \frac{1}{2} v_B \]

### Step 5: Substitute the value of \( v_A \)
Since we know \( v_A = 3v \):

\[ 3v = \frac{1}{2} v_B \]

### Step 6: Solve for \( v_B \)
To find \( v_B \), we rearrange the equation:

\[ v_B = 6v \]

### Final Answer
The speed of satellite B is \( 6v \).

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